The dynamics of dendritic growth of a crystal in an undercooled melt isdetermined by macroscopic diffusion-convection of heat and by capillary forcesacting on the nanometer scale of the solid-liquid interface width.Its modelling is useful for instance in processing techniques based on casting.The phase-field method is widely used to study evolution of such microstructural phase transformations ona continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landauequation modelling the dynamics of an order parameter determining the solid and liquid phases,including also stochastic fluctuations to obtain the qualitatively correctresult of dendritic side branching.This work presents a method to determine stochastic phase-field models from atomisticformulations by coarse-graining molecular dynamics. It has three steps: (1) a precisequantitative atomistic definition of the phase-field variable, based on the localpotential energy;(2) derivation of its coarse-graineddynamics model, from microscopic Smoluchowski molecular dynamics (that is Brownian or over damped Langevin dynamics);and (3) numerical computation of the coarse-grained model functions. The coarse-grained model approximates Gibbs ensemble averages of the atomistic phase-field, bychoosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in thisensemble average.

In this paper we introduce and analyze new mixed finite volume methods for second order elliptic problemswhich are based on H(div)-conforming approximations for the vector variable anddiscontinuous approximations for the scalar variable.The discretization is fulfilled by combining the ideas of the traditional finite volume box method andthe local discontinuous Galerkin method.We propose two different types of methods, called Methods I and II, and show that they have distinct advantagesover the mixed methods used previously.In particular, a clever elimination of the vector variable leads to a primal formulation for the scalar variablewhich closely resembles discontinuous finite element methods.We establish error estimates for these methods that are optimal for the scalar variable in both methodsand for the vector variable in Method II.

We present here a theoretical study of eigenmodes inaxisymmetric elastic layers.The mathematical modelling allows us to bring this problem to a spectral studyof a sequence of unbounded self-adjoint operators A n , $n\in \mathbb{N}$ , in a suitableHilbert space. We show that the essential spectrum of A n is an interval oftype $[\gamma,+\infty[$ and that, under certain conditions on the coefficientsof the medium, the discrete spectrum is non empty.

The accuracy of the domain embedding method from [A. Rieder, Modél. Math. Anal. Numér.32 (1998) 405-431] for the solution of Dirichlet problemssuffers under a coarse boundary approximation. To overcome this drawback the methodis furnished withan a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme.In contrast to similar, but rather theoretical, concepts already described in theliterature our approach combines a high generality with an easy-to-implement algorithm.

Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ increases on [0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $\xR_{+}^{2}$ with homogeneous Dirichlet boundary conditions by thesolution of $-\Delta u_{L}+f(u_{L})=0,$ on ]0,L[2 with adequatenon-homogeneous Dirichlet conditions. We show that the error uL - utends to zero exponentially fast, in the uniform norm.

While making informed decisions regardinginvestments in customer retention and acquisition becomes apressing managerial issue, formal models and analysis, which mayprovide insight into this topic, are still scarce. In this studywe examine two dynamic models for optimal acquisition andretention models of a monopoly, the total cost and the cost percustomer models.These models are analytically analyzed using classical, direct,methods and asymptotic expansions (for the total cost model).In order to numerically simulated the models, an innovativenumerical method was developed for solving ODE systems withinitial/final value problems.


In this paper, we consider a 2D mathematical modelling of the verticalcompaction effect in a water saturated sedimentary basin. This model isdescribed by the usual conservation laws, Darcy's law, the porosity as afunction of the vertical component of the effective stress and theKozeny-Carman tensor, taking into account fracturation effects. This modelleads to study the time discretization of a nonlinear system ofpartial differential equations. The existence is obtained by a fixed-pointargument. The uniqueness proof, by Holmgren's method, leads to work out a linear, strongly coupled, system of partial differential equations andboundary conditions.

In this paper, we are concerned with a kind of Signorini transmission problem in a unbounded domain. A variational inequality is derived when discretizing this problem by coupled FEM-BEM. To solve such variational inequality, an iterative method, which can be viewed as a variant of the D-N alternative method, will be introduced. In the iterative method, the finite element part and the boundary element part can be solved independently. It will be shown that the convergence speed of this iteration is independent of the mesh size. Besides, a combination between this method and the steepest descent method is also discussed.

The aim of this work is to present a computationally efficient algorithm to simulate thedeformations suffered by a viscoplastic body in a solidification process. This type ofproblems involves a nonlinearity due to the considered thermo-elastic-viscoplastic law. Inour previous papers, this difficulty has been solved by means of a duality method, knownas Bermúdez–Moreno algorithm, involving a multiplier which was computed with a fixed pointalgorithm or a Newton method. In this paper, we will improve the former algorithms bymeans of a generalized duality method with variable parameters and we will presentnumerical results showing the applicability of the resultant algorithm to solidificationprocesses. Furthermore, we will describe a numerical procedure to choose a constantparameter for the Bermúdez–Moreno algorithm which gives good results when it is applied tosolidification processes.

The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.

We consider multiscale systems for which only a fine-scalemodel describing the evolution of individuals (atoms,molecules, bacteria, agents) is given, while we are interested in theevolution of the population density on coarse space and timescales. Typically, this evolution is described by a coarseFokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution ofthis Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale,individual-basedsystem. As these parameters might be space- and time-dependent, theestimation is performed in every spatial discretization point and atevery time step. If the fine-scale model is stochastic, the estimationprocedure introduces noise on the coarse level.We investigate stability conditions for this procedure in thepresence of this noise and present ananalysis of the propagation of the estimation error in the numericalsolution of the coarse Fokker-Planck equation.

We propose a Lagrangian approach to deriving energy-preserving finite difference schemes for the Euler–Lagrange partial differential equations. Noether’s theorem states that the symmetry of time translation of Lagrangians yields the energy conservation law. We introduce a unique viewpoint on this theorem: “the symmetry of time translation of Lagrangians derives the Euler–Lagrange equation and the energy conservation law, simultaneously.” The proposed method is a combination of a discrete counter part of this statement and the discrete gradient method. It is also shown that the symmetry of space translation derives momentum-preserving schemes. Finally, we discuss the existence of discrete local conservation laws.

The aim of this article is to give a regularization method for an unilateral obstacle problem with obstacle ψ and second member f, which generalizes the one established by the authors of [4] in case of null obstacle and a second member is equal to constant 1.

We prove the discrete compactness property of the edge elements of any order on a classof anisotropically refined meshes on polyhedral domains. The meshes, made up oftetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl.Sci. 21 (1998) 519–549]. They are appropriately graded nearsingular corners and edges of the polyhedron.

The main objective of this paper is to provenew necessary conditions to the existence ofKAM tori. To do so, we develop a set ofexplicit a-priori estimates for smoothsolutions of Hamilton-Jacobi equations,using a combination of methods fromviscosity solutions,KAM and Aubry-Mather theories.These estimatesare validin anyspace dimension, and can be checked numericallyto detect gaps between KAM tori and Aubry-Mather sets.We apply these results to detect non-integrable regions in several examples such as a forced pendulum, two coupled penduli, andthe double pendulum.

We use the work of Milton, Seppecher, and Bouchitté on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.

In this paper, we consider C 1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that it preserves the energy and the volume.

The aim of this paper is to analyze a low order finite element methodfor a stiffened plate. The plate is modeled by Reissner-Mindlinequations and the stiffener by Timoshenko beams equations. Theresulting problem is shown to be well posed. In the case of concentricstiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysisand discretization of the first one is straightforward. The second oneis shown to have a solution bounded above and below independently of thethickness of the plate. A discretization based on DL3 finite elementscombined with ad-hoc elements for the stiffener is proposed.Optimal order error estimates are proved for displacements, rotationsand shear stresses for the plate and the stiffener. Numerical tests arereported in order to assess the performance of the method. Thesenumerical computations demonstrate that the error estimates areindependent of the thickness, providing a numerical evidence that themethod is locking-free.


We describe a constructive algorithm for obtaining smoothsolutions of a nonlinear, nonhyperbolic pair of balance lawsmodeling incompressible two-phase flow in one space dimension andtime. Solutions are found as stationary solutions of a relatedhyperbolic system, based on the introduction of an artificial timevariable.As may be expected for such nonhyperbolic systems, in general thesolutions obtained do not satisfy both components of the giveninitial data. This deficiency may be overcome, however, byintroducing an alternative “solution" satisfying both componentsof the initial data and an approximate form of a correspondinglinearized system.

In this paper, we discuss an hp-discontinuous Galerkin finiteelement method (hp-DGFEM) for the laser surface hardening ofsteel, which is a constrained optimal control problem governed by asystem of differential equations, consisting of an ordinarydifferential equation for austenite formation and a semi-linearparabolic differential equation for temperature evolution. The spacediscretization of the state variable is done using an hp-DGFEM,time and control discretizations are based on a discontinuousGalerkin method. A priori error estimates are developed atdifferent discretization levels. Numerical experimentspresented justify the theoretical order of convergence obtained.